Harnack inequality for quasiiinear elliptic equations on Riemannian manifolds

Abstract

We study viscosity solutions to degenerate and singular elliptic equations L-F vertical bar u vertical bar : = div(F'(del vertical bar del u vertical bar)/vertical bar del u vertical bar u) = h of p-Laplacian type on Riemannian manifolds, where an even function F is an element of C-1 (R) boolean AND C-2 (0, infinity) is supposed to be strictly convex on (0, infinity). Under the assumption that either F is an element of C-2 (R) or its convex conjugate F* is an element of C-2 (R) with some structural condition, we establish a (locally) uniform ABP type estimate and the Krylov-Safonov type Harnack inequality on Riemannian manifolds with the use of an intrinsic geometric quantity to the operator. Here, the C-2-regularities of F and F* account for degenerate and singular operators, respectively. (c) 2017 Elsevier Inc. All rights reserved.